On the negative limit of viscosity solutions for discounted Hamilton-Jacobi equations

Abstract

Suppose $M$ is a closed Riemannian manifold. For a $C^2$ generic (in the sense of Mañé) Tonelli Hamiltonian $H:T^{\ast }M\to \mathbb{R}$, the minimal viscosity solution $u_{\lambda }^{-}:M\to \mathbb{R}$ of the negative discounted equation \[-λu+H(x,d_xu)=c(H),\quad x\in M,\ λ>0 \] with the Mañé's critical value $c(H)$ converges to a uniquely established viscosity solution $u_0^{-}$ of the critical Hamilton-Jacobi equation \[ H(x,d_x u)=c(H),\quad x\in M \] as $\lambda \to 0_{+}$. We also propose a dynamical interpretation of $u_0^{-}$.